Non-Linear Galactic Dynamics: Action-Derived Equation and Structural Bounds

We derive the exact equation of motion for the RCD coherence field from the full disformal action in the quasi-static galactic regime. We prove a No-Go theorem: the minimal disformal action with potential V = alpha*C^2*(1-C)^2 produces field gradients Delta_C < 1e-6 across realistic galactic halos, yielding Delta_g << a_0. We classify two minimal extensions capable of restoring galactic phenomenology while preserving cosmological viability: running couplings (Completion A, 2 parameters) and conformal matter coupling (Completion B, 1 parameter). Completion B is developed into kinetic completion B' in Paper VII.